L∞/ L2 Hankel Norm Analysis and Characterization of Critical Instants for Continuous-Time Linear Periodically Time-Varying Systems

This paper is concerned with the Hankel norm analysis of linear periodically time-varying systems. An arbitrary
$\Theta\in[0,\ h)$
is first taken as the instant separating past and future, where
$h$
denotes the period of such systems, and what is called the quasi
$L_{\infty}/L_{2}$
Hankel norm at
$\Theta$
is defined. Then, a computation method of this norm for each
$\Theta$
is derived. The supremum of the quasi
$L_{\infty}/L_{2}$
Hankel norms over
$\Theta\in[0,\ h)$
is further defined as the
$L_{\infty}/L_{2}$
Hankel norm, and it is also shown that it can be computed directly without dealing with any quasi
$L_{\infty}/L_{2}$
Hankel norms. A relevant question of whether the supremum is attained as the maximum is also studied. In particular, it is discussed when and how the existence/absence of a critical instant attaining the maximum (and all the values of critical instants, if one exists) can be determined without computing all (or any of) the quasi
$L_{\infty}/L_{2}$
Hankel norms over
$\Theta\in[0,\ h)$
.